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Probabalistic interpretation of prime number theorem

Suppose there is a function f(x) which is the "probability" that the integer x is prime. The integer x is prime with probability f(x), and then divides the larger integers with probability 1/x; so as x changes from x to x+1, f(x) changes to (roughly)

f(x).(1-f(x)/x).


How do I show that? I can go on to show

df/dx +f^2/x=0

and thus 1/(logx +c) is solution but I can't show that step on how f(x) changes. please advise